Finite Group Databasebeta

← #28348388

Subgroups — #28348388

Order 13696.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293343C211yesmaximal, characteristic, complemented
2#20293280C211yesmaximal, characteristic, complemented
3#20292967C212yesmaximal, complemented
4#20292914C211yesmaximal, characteristic, complemented
5#20292859C212yesmaximal, complemented
6#20292829C211yesmaximal, characteristic, complemented
7#20292775C212yesmaximal, complemented
8C4×D856C211yesmaximal, characteristic, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#20292622C212yesmaximal, complemented
11#487×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133805C2212yes—
14C2×D856C2211yescharacteristic
15C2×C8×D107C2212yes—
16#16812098—22no—
17#16812094—22no—
18#16812085C2212yes—
19#16812084—21no—
20#16812084—21no—
21#16812084C2212yes—
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812067—21no—
26#16812067C2212yes—
27C4⋊C4×D107—21no—
28#16812043C2211yescharacteristic
29#16812040C2212yes—
30C4⋊(C107⋊C8)C2211yescharacteristic
31#16812033C2212yes—
32#16812031C2211yescharacteristic
33#16812031C2211yescharacteristic
34#16812029C2212yes—
35(C2×D107)⋊C8C2212yes—
36C4⋊Q8×C107C2211yescharacteristic
37C4⋊Q8×C107C2211yescharacteristic
38#103×C107C2212yes—
39#103×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133819—24no—
42#13133819—42no—
43C2×D428—22no—
44C2×D428—22no—
45C2×D428D411yescharacteristic
46C2×D428C2311yescharacteristic
47C2×D428C2311yescharacteristic
48C2×C4×D107—21no—
49C2×C4×D107—21no—
50C2×C4×D107—22no—
51C2×C4×D107C2312yes—
52#13133805—42no—
53D856—22no—
54C2×C107⋊C8C2312yes—
55C8×D107—42no—
56C4⋊Dic107D411yescharacteristic
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—21no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4D412yes—
66C4×Dic107—22no—
67C2×Q8×C107—21no—
68C2×Q8×C107C2312yes—
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2312yes—
75C42×C107C2311yescharacteristic
76C22×D107—21no—
77C22×D107—21no—
78C22×D107C2×D412yes—
79Q8×C107—22no—
80Q8×C107—22no—
81Q8×C107—81no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107C2×D412yes—
90D428—22no—
91D428—21no—
92D428—21no—
93D428—21no—
94D428—44no—
95D428—82no—
96D428#4112yes—
97C4×D107—22no—
98C4×D107—22no—
99C4×D107—44no—
100C4×D107—42no—
101C4×D107—42no—
102C8×C107—21no—
103C8×C107—21no—
104C107⋊C8—22no—
105C22×C107C22×D411yescharacteristic
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—22no—
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—21no—
116C4×C107—42no—
117C4×C107—42no—
118C4×C107#13512yes—
119C4×C107C2×#4111yescharacteristic
120C4×C107H(5,𝔽2)11yescharacteristic
121Dic107—22no—
122Dic107—41no—
123Dic107—41no—
124C2×C107#54611yescharacteristic
125C2×C107C2×#13511yescharacteristic
126C2×C107#57711yescharacteristic
127D107—42no—
128D107—42no—
129D107—81no—
130#3303—1071nomaximal, Sylow
131C107#330311yescharacteristic, complemented, Sylow
132#555—1071no—
133#547—1071no—
134#487D10711yescharacteristic, complemented
135#485—1072no—
136#476—1071no—
137#465—1072no—
138#461—1071no—
139#449—1072no—
140C4×D8—1071no—
141C8×D4—1071no—
142#416—1072no—
143C2×#41—1072no—
144C2×SD16—1072no—
145C2×D8—1071no—
146C22×C8—1072no—
147C4⋊Q8C2×D10711yescharacteristic
148C4⋊Q8C2×D10711yescharacteristic
149#124—2142no—
150#123—2142no—
151C22⋊Q8—1072no—
152#121—1072no—
153#121—2141no—
154#121—2141no—
155C4×D4—1071no—
156C4×D4—1071no—
157C4×D4—1071no—
158#117—1072no—
159#117—2141no—
160C2×C4⋊C4—2141no—
161#107—1071no—
162#106—1072no—
163C4⋊C8—1071no—
164#103—1072no—
165#103C2×D10712yes—
166#103C2×D10712yes—
167#102—1071no—
168#102—1071no—
169#102—1072no—
170C22⋊C8—1072no—
171C4×C8C2×D10711yescharacteristic
172#41—4282no—
173#41—2144no—
174C2×Q8—21no—
175C2×Q8C22×D10712yes—
176C2×D4—1071no—
177C2×D4—1071no—
178C2×D4—1071no—
179C2×D4—2142no—
180C2×D4—2142no—
181C22×C4—1072no—
182C22×C4—2142no—
183C22×C4—2141no—
184C22×C4—2141no—
185SD16—4282no—
186D8—2142no—
187C2×C8—1072no—
188C2×C8—4282no—
189C2×C8C22×D10711yescharacteristic
190C2×C8C22×D10711yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—1071no—
195C4⋊C4—1071no—
196C4⋊C4—1071no—
197C4⋊C4—2142no—
198C4⋊C4C22×D10712yes—
199C22⋊C4—1072no—
200C22⋊C4—2142no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C42—2142no—
206C42C22×D10711yescharacteristic
207C23—1072no—
208C23—2141no—
209C23—2141no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—8562no—
214D4—1072no—
215D4—4284no—
216D4—2142no—
217D4—2141no—
218D4—2141no—
219D4—2141no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—1072no—
223C2×C4—4284no—
224C2×C4—4282no—
225C2×C4—4282no—
226C2×C4—2141no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4D4×D10711yescharacteristic
231C2×C4D4×D10711yescharacteristic
232C2×C4C23×D10711yescharacteristic
233C8—21no—
234C8—21no—
235C8—2142no—
236C22—4281no—
237C22—4281no—
238C22—4281no—
239C22—4281no—
240C22—4281no—
241C22—2142no—
242C22—2142no—
243C22—2142no—
244C22—2142no—
245C22C2×D4×D10711yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—4281no—
250C4—4281no—
251C4—2142no—
252C4#1681213212yes—
253C4#41×D10711yescharacteristic
254C4#1681215511yescharacteristic
255C2—8561no—
256C2—4282no—
257C2—4282no—
258C2#2029327011yescharacteristic
259C2C2×#1681213211yescharacteristic
260C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#487×C107C2
socleC22×C107C22×D4
hypercenter#487D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20293343C2semidirect
2. #20293280C2semidirect
3. #20292967C2semidirect
4. #20292914C2semidirect
5. #20292859C2semidirect
6. #20292829C2semidirect
7. #20292775C2semidirect
8. C4×D856C2semidirect
9. C8×D428C2semidirect
10. #20292622C2semidirect
11. #487×C107C2semidirect
131. C107#3303semidirect
134. #487D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202933432
#202932802
#202929672
#202929142
#202928592
#202928292
#202927752
C4×D8562
C8×D4282
#202926222
#487×C1072
#3303107